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Question: Dimensional formula of Farad is \(\begin{aligned} & \text{A}\text{. }\left[ {{M}^{-1}}{{L}^{-2...

Dimensional formula of Farad is
A. [M−1L−2TQ] B. [M−1L−2T2Q2] C. [M−1L−2TQ2] D. [M−1L−2T2Q] \begin{aligned} & \text{A}\text{. }\left[ {{M}^{-1}}{{L}^{-2}}TQ \right] \\\ & \text{B}\text{. }\left[ {{M}^{-1}}{{L}^{-2}}{{T}^{2}}{{Q}^{2}} \right] \\\ & \text{C}\text{. }\left[ {{M}^{-1}}{{L}^{-2}}T{{Q}^{2}} \right] \\\ & \text{D}\text{. }\left[ {{M}^{-1}}{{L}^{-2}}{{T}^{2}}Q \right] \\\ \end{aligned}

Explanation

Solution

The formula of any quantity contains some basic formulas, like force, acceleration, work etc. To find the dimensional formula of any physical quantity, we need to first break its formula down into fundamental units, then by putting dimensional formulas for these fundamental units, we can find dimensional formulas of any quantity.

Formula used:
(a) Capacitance C=QVC=\dfrac{Q}{V}
(b) Work – W = QV (work done to bring a charge Q from infinite to some point in electric field)
(c) Work – W = Fd[ML2T−2]Fd\left[ M{{L}^{2}}{{T}^{-2}} \right](It is a mechanical work required to move a body to distance d, where applied force on it is F)

Complete step-by-step answer:
We know that Farad is the unit of capacitance, whose formula is given by
C=QVC=\dfrac{Q}{V}
Where,
Q is the charge on the capacitor’s plate and V is the electric potential difference.
We also know that the unit of charge is Q, so we need to find a unit for V.
Electric potential is defined as
V=WQV=\dfrac{W}{Q}
If we put this into capacitance formula we have,
C=Q2WC=\dfrac{{{Q}^{2}}}{W}
We know that work’s dimension is
[ML2T−2]\left[ M{{L}^{2}}{{T}^{-2}} \right], we can put this into capacitance (C) formula as;
So, the dimension of C will be
Q2ML2T−2\dfrac{{{Q}^{2}}}{M{{L}^{2}}{{T}^{-2}}}
Which can be further simplified as;
[Q2M−1L−2T2]\left[ {{Q}^{2}}{{M}^{-1}}{{L}^{-2}}{{T}^{2}} \right]
So, the correct answer is B.

Note: Any physical quantity can be expressed using 7 fundamental units. These fundamental units are: Mass (M), Length (L), Time (T), Electric current (I), Temperature (θ\theta ), Charge (Q) and Luminous intensity (J).
Dimension formula for a physical quantity is a formula which tells us how and which fundamental units have been used in the measurement of that quantity.